direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D5⋊M4(2), D10⋊11M4(2), Dic5.16C24, C5⋊C8⋊2C23, D5⋊(C2×M4(2)), D5⋊C8⋊9C22, C2.5(C23×F5), C10⋊2(C2×M4(2)), C10.3(C23×C4), C4.F5⋊13C22, C5⋊2(C22×M4(2)), C23.53(C2×F5), (C22×C4).25F5, C4.58(C22×F5), C20.98(C22×C4), (C22×C20).29C4, (C23×D5).19C4, (C4×D5).91C23, C22.F5⋊7C22, D10.45(C22×C4), C22.19(C22×F5), Dic5.45(C22×C4), (C2×Dic5).363C23, (C22×Dic5).283C22, (C2×C4×D5).40C4, (C2×D5⋊C8)⋊13C2, (C2×C5⋊C8)⋊10C22, (C2×C4.F5)⋊14C2, (C4×D5).97(C2×C4), (C2×C4).173(C2×F5), (D5×C22×C4).32C2, (C2×C20).152(C2×C4), (C2×C4×D5).405C22, (C2×C22.F5)⋊12C2, (C2×C10).97(C22×C4), (C22×C10).79(C2×C4), (C2×Dic5).199(C2×C4), (C22×D5).133(C2×C4), SmallGroup(320,1589)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 906 in 298 conjugacy classes, 148 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×4], C22, C22 [×2], C22 [×20], C5, C8 [×8], C2×C4 [×2], C2×C4 [×4], C2×C4 [×22], C23, C23 [×10], D5 [×4], D5 [×2], C10, C10 [×2], C10 [×2], C2×C8 [×12], M4(2) [×16], C22×C4, C22×C4 [×13], C24, Dic5 [×2], Dic5 [×2], C20 [×4], D10 [×8], D10 [×10], C2×C10, C2×C10 [×2], C2×C10 [×2], C22×C8 [×2], C2×M4(2) [×12], C23×C4, C5⋊C8 [×8], C4×D5 [×16], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×D5 [×2], C22×D5 [×4], C22×D5 [×4], C22×C10, C22×M4(2), D5⋊C8 [×8], C4.F5 [×8], C2×C5⋊C8 [×4], C22.F5 [×8], C2×C4×D5 [×4], C2×C4×D5 [×8], C22×Dic5, C22×C20, C23×D5, C2×D5⋊C8 [×2], C2×C4.F5 [×2], D5⋊M4(2) [×8], C2×C22.F5 [×2], D5×C22×C4, C2×D5⋊M4(2)
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], M4(2) [×4], C22×C4 [×14], C24, F5, C2×M4(2) [×6], C23×C4, C2×F5 [×7], C22×M4(2), C22×F5 [×7], D5⋊M4(2) [×2], C23×F5, C2×D5⋊M4(2)
Generators and relations
G = < a,b,c,d,e | a2=b5=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=b3, be=eb, dcd-1=b2c, ce=ec, ede=d5 >
►On 80 points
(1 69)(2 70)(3 71)(4 72)(5 65)(6 66)(7 67)(8 68)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 73)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 33)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 57)(56 58)
(1 58 75 31 9)(2 32 59 10 76)(3 11 25 77 60)(4 78 12 61 26)(5 62 79 27 13)(6 28 63 14 80)(7 15 29 73 64)(8 74 16 57 30)(17 41 55 39 68)(18 40 42 69 56)(19 70 33 49 43)(20 50 71 44 34)(21 45 51 35 72)(22 36 46 65 52)(23 66 37 53 47)(24 54 67 48 38)
(1 13)(2 80)(3 64)(4 30)(5 9)(6 76)(7 60)(8 26)(10 28)(11 73)(12 16)(14 32)(15 77)(17 51)(18 22)(19 66)(20 48)(21 55)(23 70)(24 44)(25 29)(27 58)(31 62)(33 47)(34 38)(35 68)(36 56)(37 43)(39 72)(40 52)(41 45)(42 65)(46 69)(49 53)(50 67)(54 71)(57 78)(59 63)(61 74)(75 79)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)
(1 65)(2 70)(3 67)(4 72)(5 69)(6 66)(7 71)(8 68)(9 46)(10 43)(11 48)(12 45)(13 42)(14 47)(15 44)(16 41)(17 74)(18 79)(19 76)(20 73)(21 78)(22 75)(23 80)(24 77)(25 38)(26 35)(27 40)(28 37)(29 34)(30 39)(31 36)(32 33)(49 59)(50 64)(51 61)(52 58)(53 63)(54 60)(55 57)(56 62)
G:=sub<Sym(80)| (1,69)(2,70)(3,71)(4,72)(5,65)(6,66)(7,67)(8,68)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,73)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,33)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,57)(56,58), (1,58,75,31,9)(2,32,59,10,76)(3,11,25,77,60)(4,78,12,61,26)(5,62,79,27,13)(6,28,63,14,80)(7,15,29,73,64)(8,74,16,57,30)(17,41,55,39,68)(18,40,42,69,56)(19,70,33,49,43)(20,50,71,44,34)(21,45,51,35,72)(22,36,46,65,52)(23,66,37,53,47)(24,54,67,48,38), (1,13)(2,80)(3,64)(4,30)(5,9)(6,76)(7,60)(8,26)(10,28)(11,73)(12,16)(14,32)(15,77)(17,51)(18,22)(19,66)(20,48)(21,55)(23,70)(24,44)(25,29)(27,58)(31,62)(33,47)(34,38)(35,68)(36,56)(37,43)(39,72)(40,52)(41,45)(42,65)(46,69)(49,53)(50,67)(54,71)(57,78)(59,63)(61,74)(75,79), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,65)(2,70)(3,67)(4,72)(5,69)(6,66)(7,71)(8,68)(9,46)(10,43)(11,48)(12,45)(13,42)(14,47)(15,44)(16,41)(17,74)(18,79)(19,76)(20,73)(21,78)(22,75)(23,80)(24,77)(25,38)(26,35)(27,40)(28,37)(29,34)(30,39)(31,36)(32,33)(49,59)(50,64)(51,61)(52,58)(53,63)(54,60)(55,57)(56,62)>;
G:=Group( (1,69)(2,70)(3,71)(4,72)(5,65)(6,66)(7,67)(8,68)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,73)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,33)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,57)(56,58), (1,58,75,31,9)(2,32,59,10,76)(3,11,25,77,60)(4,78,12,61,26)(5,62,79,27,13)(6,28,63,14,80)(7,15,29,73,64)(8,74,16,57,30)(17,41,55,39,68)(18,40,42,69,56)(19,70,33,49,43)(20,50,71,44,34)(21,45,51,35,72)(22,36,46,65,52)(23,66,37,53,47)(24,54,67,48,38), (1,13)(2,80)(3,64)(4,30)(5,9)(6,76)(7,60)(8,26)(10,28)(11,73)(12,16)(14,32)(15,77)(17,51)(18,22)(19,66)(20,48)(21,55)(23,70)(24,44)(25,29)(27,58)(31,62)(33,47)(34,38)(35,68)(36,56)(37,43)(39,72)(40,52)(41,45)(42,65)(46,69)(49,53)(50,67)(54,71)(57,78)(59,63)(61,74)(75,79), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,65)(2,70)(3,67)(4,72)(5,69)(6,66)(7,71)(8,68)(9,46)(10,43)(11,48)(12,45)(13,42)(14,47)(15,44)(16,41)(17,74)(18,79)(19,76)(20,73)(21,78)(22,75)(23,80)(24,77)(25,38)(26,35)(27,40)(28,37)(29,34)(30,39)(31,36)(32,33)(49,59)(50,64)(51,61)(52,58)(53,63)(54,60)(55,57)(56,62) );
G=PermutationGroup([(1,69),(2,70),(3,71),(4,72),(5,65),(6,66),(7,67),(8,68),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(17,74),(18,75),(19,76),(20,77),(21,78),(22,79),(23,80),(24,73),(25,34),(26,35),(27,36),(28,37),(29,38),(30,39),(31,40),(32,33),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(55,57),(56,58)], [(1,58,75,31,9),(2,32,59,10,76),(3,11,25,77,60),(4,78,12,61,26),(5,62,79,27,13),(6,28,63,14,80),(7,15,29,73,64),(8,74,16,57,30),(17,41,55,39,68),(18,40,42,69,56),(19,70,33,49,43),(20,50,71,44,34),(21,45,51,35,72),(22,36,46,65,52),(23,66,37,53,47),(24,54,67,48,38)], [(1,13),(2,80),(3,64),(4,30),(5,9),(6,76),(7,60),(8,26),(10,28),(11,73),(12,16),(14,32),(15,77),(17,51),(18,22),(19,66),(20,48),(21,55),(23,70),(24,44),(25,29),(27,58),(31,62),(33,47),(34,38),(35,68),(36,56),(37,43),(39,72),(40,52),(41,45),(42,65),(46,69),(49,53),(50,67),(54,71),(57,78),(59,63),(61,74),(75,79)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80)], [(1,65),(2,70),(3,67),(4,72),(5,69),(6,66),(7,71),(8,68),(9,46),(10,43),(11,48),(12,45),(13,42),(14,47),(15,44),(16,41),(17,74),(18,79),(19,76),(20,73),(21,78),(22,75),(23,80),(24,77),(25,38),(26,35),(27,40),(28,37),(29,34),(30,39),(31,36),(32,33),(49,59),(50,64),(51,61),(52,58),(53,63),(54,60),(55,57),(56,62)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 35 |
| 0 | 0 | 0 | 0 | 6 | 35 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 1 |
| 0 | 0 | 0 | 0 | 6 | 35 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 32 | 28 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,6,0,0,0,0,0,0,40,6,0,0,0,0,35,35],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,1,0,0,0,0,6,6,0,0,0,0,0,0,6,6,0,0,0,0,1,35],[40,4,0,0,0,0,2,1,0,0,0,0,0,0,0,0,32,0,0,0,0,0,28,9,0,0,40,0,0,0,0,0,0,40,0,0],[40,40,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5 | 8A | ··· | 8P | 10A | ··· | 10G | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | M4(2) | F5 | C2×F5 | C2×F5 | D5⋊M4(2) |
| kernel | C2×D5⋊M4(2) | C2×D5⋊C8 | C2×C4.F5 | D5⋊M4(2) | C2×C22.F5 | D5×C22×C4 | C2×C4×D5 | C22×C20 | C23×D5 | D10 | C22×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 8 | 2 | 1 | 12 | 2 | 2 | 8 | 1 | 6 | 1 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_5\rtimes M_{4(2)} % in TeX
G:=Group("C2xD5:M4(2)"); // GroupNames label
G:=SmallGroup(320,1589);
// by ID
G=gap.SmallGroup(320,1589);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,1123,102,6278,818]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=b^3,b*e=e*b,d*c*d^-1=b^2*c,c*e=e*c,e*d*e=d^5>;
// generators/relations